2005/11/30 by V. Araujo, Vitor Araujo, M. Pacifico +5 · 102 citations
Engineering · Mathematics · Physics and Astronomy · #Attractor #Differentiable function #Ergodic theory #Flow (mathematics) #Homoclinic orbit #Jacobian matrix and determinant #Lebesgue measure #Lorenz system #Mathematical Dynamics and Fractals #Measure (data warehouse) #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations #math.DS #msc:37C10 #msc:37C40 #msc:37D30
paper · pdf · doi:10.1090/s0002-9947-08-04595-9
published in Transactions of the American Mathematical Society 361(5), 2431-2485 (American Mathematical Society) · 55 pages, extra figures (now a total of 16), major rearrangement of sections and corrected proofs, improved introduction
arxiv created 2007/03/22 · openalex publication_date 2008/12/03 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We prove that a singular-hyperbolic attractor of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with time reparametrization, then their orbits coincide. Second, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Gibbs state and is an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows. In particular these results can be applied (i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz flows, (iii) to the attractors appearing in the unfolding of certain resonant double homoclinic loops, (iv) in the unfolding of certain singular cycles and (v) in some geometrical models which are singular-hyperbolic but of a different topological type from the geometric Lorenz models. In all these cases the results show that these attractors are expansive and have physical measures which are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Gibbs states.